Flight of the Golf Ball

Now we have launched the golf ball, with some ball speed, launch angle,
and spin. On this page, we will see how that turns into a ball flight,
with trajectory shape and distance.

This is very much affected by the air through which the golf
ball
flows. We all know the air does some bad things to our golf shots. It
turns sidespin into slices and hooks, and air resistance -- or "drag"
-- slows down the ball and steals distance. What most people don't know
is that air provided a lot of that distance in the first place.
In a vaccuum, the ball would not go nearly as far.

Let's take a look at why that should be.
The pictures are take from a web article on Golf
Ball Aerodynamics,
by Steve Aoyama of Titleist. I won't go into the aerodynamics here, but
there's plenty of good articles on the Internet, starting with this
one. For more, just search on [golf ball aerodynamics];
it will turn up pages of them.

The first picture shows air flowing past a non-spinning ball.
The air flows to the right, but that is the same aerodynamically as the
ball moving to the left through the air. Think of it as a picture of
the moving ball, with the camera moving along with the ball. A few ways
to look at
this:

In
terms of forces, the wind blows the ball to the right. Since the ball
is moving to the left, the wind slows the ball down. The aerodynamic
force slowing the ball is called "drag".

In terms
of energy, the air has to deflect around the ball
and come back together after the ball has passed. It takes energy to
move the air around like that. That energy is transferred from the
moving ball to the air, so the ball slows down.

Now look at the second picture. This time, the ball is spinning
clockwise in the left-to-right flow of air, corresponding to backspin.
The effect of the spin is to deflect the air downward as it streams
past the ball. So, in addition to disturbing the airflow, the ball is
pressing the air down.

What would Newton have to say about this? "Equal and opposite
reaction." The ball exerts a force on the air to push it down. The
reaction is an equal force that the air applies to the ball, pushing it
up. This force, shown as a red arrow in the picture, is called "lift".
The lift force has a magnitude and a direction.

The
direction is always at right angles to the direction the ball is
moving, and is in the plane that the ball is spinning.

The
magnitude increases with ball speed, with the spin, with the density of
the air, and depends on some aerodynamic properties of the ball's
surface -- like the dimples.

The
result is a collection of forces as the ball moves through the air. The
ball's motion is described by its speed, direction, and spin at any
point along its path. The forces are:

Drag,
in exactly the opposite direction to the one the ball is moving.

Weight, straight down.

Lift,
perpendicular to the path of the ball and in the plane of the spin.

Note that I did not say that lift is an upwards
force. It is at
right angles to the path of the ball, and in the plane of spin. Here
are a couple of interesting results of that distinction:

The
spin plane doesn't have to be straight up and down. In this picture,
the
spin plane is inclined a little to the left, so the lift force (black
arrow) is also inclined to the left. Therefore, it has components (gray
arrows) upward and leftward. Yes, lift still produces an upward force.
But it also has a force to the left. This will be a centripetal force,
which will curve the flight of the ball to the left... a hook for a
right-handed golfer.

So the spin on the ball is a combination of mostly backspin but some
sidespin as well, and that means that the resulting lift force is a
combination of upwards force keeping the ball in the air and hook or
slice force curving the ball left or right.

During
the time that the ball is climbing, the path of the ball is tilted
upward. Since the lift force has to be perpendicular to the path of the
ball, it is tilted "backward". That is, the lift has an upward
component keeping the ball in the air, and a backward component
decelerating the ball from it's progress down the fairway. So the ball
slows down during its climb, not just from drag but also due to the
backward tilt of the lift.

From force to trajectory

So air is our enemy (drag) but also our friend (lift). Lift keeps the
ball in the air longer, giving us more distance. Which brings us to a
question I'm asked all the time: is there some simple formula
for distance?
The answer is that there are several, but none are very good. In
particular, any simple formula that includes loft is too simplistic to
reflect
the complex actions of aerodynamics, especially for the longer clubs.

The way trajectories are calculated are not by plugging
numbers into a simple formula. Rather, a computer program
is used to compute the forces on the ball and, a few inches or feet at
a time, move the ball through its trajectory. Such programs aren't
limited to the club designer's (or clubfitter's) personal computer; the
same
kind of program is used in launch monitors and golf simulators.

I wasn't planning on going into how trajectory programs work.
But then,
I realized that their working gives insight into how trajectories
themselves work -- how the physics turns the forces into a trajectory.
So here goes; here's how those programs work:

Start at the instant of launch, knowing the
launch conditions.

The program starts
with the speed, direction, and spin of the
ball at that instant. From that the program can figure -- or
already knows:

The three forces on
the ball: lift, drag, and weight.

The
acceleration of the ball, just by applying F=ma.
This is acceleration in three dimensions, so it includes up-down and
left-right acceleration, as well as downrange acceleration.

The program steps forward a tiny increment in
time, probably something between 1/100 and 1/10 of a second. That
involves:

Moving the ball in space,
in the direction of the current
velocity. For instance, if the velocity is 200 feet per second and the
interval is 1/100 of a second, then the ball moves 2 feet in the
direction the ball was already headed. (That's 200 feet per second
times 1/100 of a second.)

Computing a
new velocity. We have an old velocity, an
acceleration (which may not be in the same direction in the old
velocity -- but we know
how to handle that),
and a time interval over which the acceleration acts. So it's easy to
compute the new velocity. The new velocity consists of a new speed and
a new direction, both slightly different from the old ones.

Computing a new spin. The spin just loses a small percentage
of its magnitude every time interval, due to air resistance.

The program now has a new set of conditions:
speed, direction,
spin -- and a new position. Go back to step #2 with the new information
and repeat. Continue this loop until the ball's height is zero; that
means it has hit the ground.

The remarkable thing is that this is how the real world
operates, too.
It does so continuously, not a step at a time. But it works by
continually looking at the velocity, turning that into forces using F=ma,
and changing that velocity according to the acceleration.

Spin, lift, and ballooning

As we said, lift is our friend. Right? Well, not always.
There can
be too much of a good thing. For instance, you've heard about a ball
losing distance because it "balloons". That is what comes from an
overabundance of lift. Let's see this in more detail. (The studies
below were done using trajectory
programs from Max Dupilka and Tom Wishon. I needed both
programs, because each is better than the other at some job.)

Here
are some trajectories for a pretty big hitter. With a good driver
swing, he gets a 165mph ball speed (that comes from good impact at
110mph clubhead speed) and a 10° launch angle. Let's see what
happens as we vary the spin of the ball.

First
of all, with no spin he gets no distance. If you're
going to launch the ball with a driver, the angle is low enough that
you really really need lift to keep the ball in
the air. With no spin, the ball flies less than half it's potentional
down the fairway.

Even a little spin (1000rpm is
pretty low spin for a hard
hit with a driver) provides enough lift to go twice as far as the ball
did with none at all.

As the spin increases, so
does the distance. At least up to
2800rpm. But that's the best distance he's going to get. (True for
165mph and10°; the ideal spin will be different for every ball
speed and launch angle.)

Any further increase in
spin produces no more distance. By
6000rpm, the distance is back almost to where it was at 1000rpm. This
effect is called "ballooning", and is due to too much lift.

Look at the trajectory at 6000rpm. It curves upward markedly,
continuing at an even higher angle than the ball was originally
launched. (This also happens, but much more modestly, with the ideal
spin.) Remember what we saw earlier; while the ball is climbing, some
of the lift force holds the ball back. That's why lift can be too much
of a good thing. It is holding the ball back and
causing the
trajectory to be steeper, which holds the ball back even more. The
result is that, even though the ball stays in the air longer, the
travel time is spent climbing, not getting downrange.

The ball
designer can adjust the lift and drag
of the ball, just by playing with the dimples on the ball's surface.
Changing the area, depth, and pattern of the dimples can change the
lift-to-drag ratio by a ratio of 3 to 1. The graph below shows
trajectories
for the same launch conditions, but varying the lift-to-drag ratio over
a much wider range than you could accomplish with dimple pattern alone.

For a golfer with 110mph of clubhead speed who has a 10°
driver, a
little extra lift is not a bad thing. True, a bit of ballooning is
visibly evident on the graph as the lift increases. But, until the lift
gets to more than 1.4 times what is "normal" for a median-lift ball,
the result is more carry distance. Only when lift exceeds about 1.5
times normal do we see ballooning hurt rather than help.

But how universal is this effect? Is the optimum lift always 1.5 time
greater than normal? We have to take it with a few grains of salt. For
instance:

10° is not necessarily
the ideal loft for a golfer with
a 110mph clubhead speed. The ideal launch
conditions for such a golfer involve a higher launch angle than this
graph indicates. If that launch angle is achieved, then the ball will
not need nearly as much spin -- and ballooning will hurt distance at
less lift than indicated here.

The higher your
clubhead speed, the less lift you need to
keep the ball in the air. So the optimum lift will not be the same for
every clubhead speed. (We'll see that below.)

In
fact, bigger hitters using drivers that give their optimum launch angle
and spin will get maximum distance will less than normal
lift. That's part of the reason why they can drive the ball farther at
high altitude. (The rest of the reason is reduced drag.)

All
the graphs on this page refer to carry only, not
total distance. But bounce and roll are hurt by higher lift, because
the ball will come down to the ground more vertically -- and possibly
more backspin. So maximum total
distance is achieved at lower lift than maximum carry is.

Numbers and
the "Launch Space"

Now we know that the price of ballooning
changes with clubhead speed. Let's look at this a little more closely,
since it is so important to clubfitting -- especially drivers. Today,
the state-of-the-art
method of fitting drivers is with launch monitors, and the thing the
launch monitors look at are, of course, the launch conditions: ball
speed, launch angle, and spin. How do these relate, when the objective
is maximum carry distance?

Engineers
and mathematicians would call this the "launch space". It is a graph
that starts with the launch parameters and plots the carry distance.
That's a four-dimensional space (ball speed, launch angle, spin, and
distance), which is hard to show on a two-dimensional page. But let's
see if we can find some useful ways to visualize it.

Here's
how carry distance varies with spin for several representative ball
speeds.
For each speed we used a good launch angle for that speed and just
varied the spin. Points to note:

The
four ball speeds used were:

100mph -
a senior or less athletic golfer, with a clubhead speed about 70mph.

130mph - this represents good impact at 85-90mph. It
corresponds to the majority of decent golfers.

160mph
- a big hitter that you might run across at your course.

190mph
- one of the bigger hitters on the pro tour, but not yet a serious
competitor in long drive championships.

The best spin for the 190mph golfer was about 2000rpm,
while the 100mph golfer did best above 3000rpm. Actually, the optimum
was probably around 4000rpm, but the distance was very close to that at
3000rpm.

For
the 190mph golfer, that 2000rpm was with a launch angle of 10°,
while the 100mph golfer had a launch angle of 20° and did best
over
300rpm.
So the effect of spin is even more than it appears. That's because the lower
launch angle needs more spin to keep the ball in
the air.

Now let's look at the other side of the picture: launch angle. Here are
the corresponding plots, using a good spin for each speed and plotting
distance against launch angle.

The
golfer with the highest ball speed does best with the lowest launch
angle: 190mph at 11°. The best launch angle for a golfer with
only
100mph ball speed is more than double that, at 23°.

As
before, we held the spin constant at a "good" value for
each ball speed. That spin varied from 2000rpm for the 190mph golfer to
3500rpm for the 100mph golfer.

One of the most startling things on this page is how high the optimum
launch angle is for most golfers. Next we'll see what's really going on.

Now
we've seen how distance varies with spin, and how it varies with launch
angle. But it's very instructive to see how distance varies with both
at the same time. Let's add a dimension to what we're looking at.
Here's a chart that shows more of the launch space --
distance vs both spin and launch angle for a ball
speed of 124mph.
(That corresponds to a well-struck ball off a driver whose head speed
is 85mph, comparable to a typical male golfer.)

Spin(rpm)

Launch Angle (Degrees)

8

10

12

14

16

18

20

22

24

26

5500

193

195

196

196

195

194

192

190

187

183

5000

193

195

197

197

197

196

194

192

189

185

4500

192

195

197

198

198

197

196

194

191

188

4000

191

195

197

199

199

199

198

196

193

190

3500

190

194

197

199

200

200

199

198

195

193

3000

187

192

196

198

200

200

200

199

197

195

2500

183

189

194

197

199

200

201

200

199

197

2000

178

185

190

195

197

199

200

200

200

198

1500

170

179

185

190

194

197

199

199

200

199

1000

159

168

177

183

188

192

195

197

198

198

Because
it is difficult to make much sense of a table of numbers, I have
color-coded it to give it some contour -- so the colors are the third
dimension. The colors go from hot (red) at
the maximum distance of 201 yards to cool (blue) for all distances
under 194 yards. While the location of the peak is pretty clear, the
shape of the space is interesting. The
peak appears to be on a "ridge" of good yardage that runs from upper
left (low launch, high spin) to lower right (high launch, low spin).

The
smaller, annotated picture at the right shows where the ridge is.
Distance falls off rapidly on either side of the ridge. At the lower
left -- low launch and low spin -- distance is lost because there isn't
enough aerodynamic lift to keep the ball in the air. At the upper right
-- high launch and high spin -- ballooning kills the distance. But
moving along the ridge, you can change launch angle and spin by pretty
large amounts with fairly little loss of distance.

"Moving
along the ridge" means you have to change both angle and spin so you
stay on the ridge. Let's see what happens if you just change one or the
other. Suppose you had a driver that launched the ball at 124mph (this
whole table assumes a ball speed of 124mph), with a 14º launch
angle and
a 4000rpm spin. The resulting distance is 199 yards. The peak distance
of 201 yards lives at 20º and 2500rpm.

Suppose
you changed the launch angle to 20º without changing the spin.
You would lose a yard, to 198 yards.

Suppose you
changed the spin to 2500rpm without changing the launch angle. You
would lose two yards, to 197 yards.

But, if you
change both, you'll pick up the two yards to the theoretical maximum.

This
is very significant in the fitting of drivers. Here is a table showing
a "family" of drivers. The design is very simple: 200g clubhead, a COR
of 0.83 (the maximum allowed by the USGA), nothing fancy about the
weight distribution, etc. Each row of the table shows a different loft;
that is all that is varied from one driver to another in the "family".
(Actually, it is effective loft at impact, a number which includes the
effect of shaft bend.)

What the table shows for each
driver is
the distance, launch angle, and spin for each driver in the family --
assuming the ball is being struck at an 85mph clubhead speed and a zero
angle of attack.

Because
we have a distance, LA, and spin for each driver, we can plot the
drivers on the launch space table we just calculated. When we do this,
we see them lying along the red dotted line. A few points worth noting
here:

Increasing loft increases both the
launch angle and
the spin. This means that it does not move along the ridge, but almost
at right angles to it.

Since it crosses the ridge
at a right
angle, there is a fairly well-defined maximum distance as you vary the
loft. And the maximum along the red line of loft variation is where it
intersects the black line of the ridge. This should not be a big
surprise when you think about the shape of the contours on the chart.

The
maximum distance along the red line is a long way from the
peak
of the launch space. The difference is 6º of launch angle
(14º to 20º), and 1500rpm of spin (4000rpm to
2500rpm). Even
so, the distance that driver can hit the ball is only two
yards short of the theoretical maximum for that ball speed.

This
last point is important! It tells us that a practical, well-fit driver
is one that lives at the top of the ridge. It doesn't have to be at the
peak, but it can still give almost as much distance as the peak. I've
plotted the same chart for clubhead and ball speeds from long-drive
champions to little old ladies from Pasadena. The simple driver that
lives on top of the ridge is never more than 2 yards short of the
theoretical maximum.

Summary of ball flight

Summarizing the lessons from this page:

Backspin creates lift, which is necessary to get
the best distance from the golf ball.

The amount
of spin you need for the best distance will vary, mostly with ball
speed, but also with launch angle.

The less ball
speed you generate, the more launch angle you need
to get distance. Everybody has heard this by now, but the best launch
angle is a lot bigger than most people realize.

The
less ball speed you generate, the more spin is your friend.

The
lower you hit the ball, the more spin is your friend. Conversely, the
higher you hit the ball, the less spin you want.

If
your driver design combines launch angle and spin to put it "on the
ridge" for your ball speed, then you are getting almost as much
distance as the maximum that physics says you can get. In this context,
"almost" means "within two yards."